In this note we collect some references that discuss the notion of connection on vector bundle

1. Differential geometry by Loring Tu
2. Geometry of Differential forms by Shigeyuki Morita
3. Global Calculus by S Ramanan
4. From Calculus to Cohomology by Madsen
5. Natural Operations in differential geometry by Kolar, Michor, Slovak
6. Foundations of Differential geometry by Kobayashi and Nomizu
7. Differential geometry by Taubes
8. Geometry of Physics by Theodore Frankel
9. Modern differential geometry for Physicists by Chris Isham

Differential Geometry

by Loring Tu

Let \(E\rightarrow M\) be a \(C^\infty\) vector bundle over \(M\). A connection on \(E\) is a map

\[\nabla:\mathfrak{X}(M)\times \Gamma(E)\rightarrow \Gamma(E)\]

such that for \(X\in \mathfrak{X}(M)\) and \(s\in \Gamma(E)\),

1. \(\nabla_Xs\) is \(\mathfrak{F}\)-linear in \(X\) and \(\mathbb{R}\)-linear in \(s\)
2. (Leibniz rule) if \(f\) is a \(C^\infty\) function on \(M\), then \(\nabla_X(fs)=X(f)s+f\nabla_Xs\)

Since \(X(f)=(df)(X)\), the Leibniz rule may be written as \(\nabla_X(fs)=(df)(X)s+f\nabla_Xs,\) or, suppressing \(X\),

\[\nabla(fs)=df\cdot s+f\nabla s\].

Geometry of Differential forms

by Shigeyuki Morita

defines Connection on a vector bundle \(E\rightarrow M\) as a map \(\nabla:\mathfrak(M)\times \Gamma(E)\rightarrow \Gamma(E)\) satisfying certain conditions

Global Calculus

by Ramanan

defines connection on a vector bundle \(E\rightarrow M\) as a splitting of associated first order symbol sequence

In the next page they give an equivalent description. This may look familiar than the "first order symbol notion"

From Calculus to Cohomology

by Madsen and Tornehave

defines connection on a vector bundle \(\xi \rightarrow M\) as an \(\mathbb{R}\)-linear map \(\nabla:\Omega^0(\xi)\rightarrow \Omega^1(M)\otimes_{\Omega^0(M)}\Omega^0(\xi)\) satisfying certain conditions

Natural Operations in differential geometry

by Kolar, Michor, Slovak

first discuss notion of connection on a fibre bundle

Then the discuss connection on principal bundle. After that they discuss connection on vector bundle, along with equivalent description using "connector \(K:TE\rightarrow E\)"

Two interesting things that appear here are the following:

1. In previous discussion we said there is no obvious map \(TE\rightarrow E\) (other than the usual projection, which is useless for us). In this case, connection is described using "connector".
2. They already discussed connection on principal bundle. Frame bundle of a vector bundle is an example of a principal bundle. They are highlighting here that, connection on a vector bundle has to come from connection on the associated principal bundle. This was a very big relief for me when I was reading connections for first time.

Foundations of differential geometry

by Kobayashi and Nomizu

unfortunately does not describe connection on vector bundle in an independent way.

In first chapter they describe the notion of a principal \(G\)-bundle.

They also discuss the notion of associated vector bundle for a principal \(G\)-bundle \(P(M,G)\) and a representation \(\rho:G\rightarrow {\rm{GL}}(\mathbb{R}^n)\).

Stating with a connection on principal bundle \(P(M,G)\), they associate "parallel displacement" on fibers of \(E\rightarrow M\). Using this they define "covariant derivative". All this data fit nicely with our notion of connection on vector bundle. But, I expected they would discuss it independently.

Differential geometry

by Taubes

As far as I understand, in this book, Taubes want to reserve the term covariant derivative for vector bundles and connections for principal bundles.

Given a principal bundle \(E\rightarrow M\), a covariant derivative is a map \(\nabla:C^\infty(M;E)\rightarrow C^\infty(M;T^*M\otimes E)\)

Some may think this version looks straightforward than that of Madesen calculus to Cohomology book definition. We are aware of differential forms taking values in vector bundle, such vector bundles are \(E\) and \(E\otimes T^*M\) and a connection is just a map \(\nabla:C^\infty(M;E)\rightarrow C^\infty(M;T^*M\otimes E)\).

Geometry of Physics

by Theodore Frankel

Definition of connection on vector bundle is written in a simple way. Not all term is clear from the definition, but, it gives an idea, with less notation.

Modern differential geometry for Physicists

by Chris Isham

As in the book of Kobayashi and Nomizu, they first define connection on principal bundles and use it to define connection on vector bundles.

Connection on principal bundle gives "parallel transpor" for fibers of associated vector bundle (we assume we already have a representation \(\rho:G\rightarrow {\rm{GL}}(\mathbb{R}^n)\). This parallel transport on vector bundle is used to define connection on vector bundle.